Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) 3], Friedman and Halilovi? (2011) 5], Friedman and Honzik (2008) 6], Friedman and Magidor (2009) 8], Friedman and Zdomskyy (2010) 10], Honzik (2010) 12]).